PseudoGap Superconductivity and Superconductor-Insulator transition

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15 Νοε 2013 (πριν από 3 χρόνια και 8 μήνες)

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PseudoGap Superconductivity and
Superconductor
-
Insulator transition

In collaboration with:


Vladimir Kravtsov ICTP Trieste


Emilio Cuevas University of Murcia


Lev Ioffe Rutgers University


Marc Mezard Orsay University


Mikhail Feigel’man

L.D.Landau Institute, Moscow

Short publication
:

Phys Rev Lett.
98
, 027001 (2007)

Superconductivity v/s Localization


Granular systems with Coulomb interaction

K.Efetov 1980 et al

Bosonic mechanism”


Coulomb
-
induced suppression of Tc in
uniform films
“Fermionic mechanism”

A.Finkelstein 1987 et al


Competition of Cooper pairing and
localization (no Coulomb)

Imry
-
Strongin,
Ma
-
Lee
, Kotliar
-
Kapitulnik,

Bulaevsky
-
Sadovsky(mid
-
80’s)

Ghosal, Randeria, Trivedi 1998
-
2001

There will be no
grains
and


no
Coulomb

in this talk !

Bosonic mechanism:



Control parameter

E
c

= e
2
/2C


Plan of the talk

1.
Motivation from experiments

2.
BCS
-
like theory for critical eigenstates


-

transition temperature


-

local order parameter

3.
Superconductivity with pseudogap


-

transition temperature v/s pseudogap



4. Quantum phase transition: Cayley tree

5. Conclusions and open problems





Example: Disorder
-
driven


S
-
I transition in TiN thin films

T.I.Baturina et al Phys.Rev.Lett

99

257003 2007


Specific Features of Direct SIT:


Insulating behaviour of the R(T) separatrix



On insulating side of SIT, low
-
temperature


resistivity is activated: R(T) ~ exp(T
0
/T)



Crossover to VRH at higher temperatures


Seen in TiN, InO, Be (extra thin)


all are

amorphous, with low electron density

There are other types of SC suppression by disorder !

Strongly insulating InO

and nearly
-
critical TiN

.

















Kowal
-
Ovadyahu 1994

Baturina et al 2007

0
2
4
6
8
10
12
14
16
18
9
10
11
12
13
14
15
16
17

I2
10
1
0.4
0.2
T [K]
0.1
0.06

ln(R[Ohm])
1/T[K]
I2: T
0

= 0.38 K

R
0

= 20 k
W

d = 5 nm

0
1
2
3
4
9
10
11
12
13
14
15
16
17

ln(R[Ohm])
1/(T[K])
1/2
d = 20 nm

T
0

= 15 K

R
0

= 20 k
W

What is the charge quantum ? Is it the same
on left

and
on right?

Giant magnetoresistance near SIT

(Samdanmurthy et al, PRL
92
, 107005 (2004)

Experimental puzzle:

Localized Cooper pairs

.







D.Shahar & Z.Ovadyahu

amorphous InO 1992

V.Gantmakher et al InO

D.Shahar et al InO

T.Baturina et al TiN

Bosonic v/s Fermionic
scenario ?


None of them is able


to describe data on


InO
x

and TiN

Major exp. data calling for a new theory


Activated resistivity

in insulating a
-
InO
x


D.Shahar
-
Z.Ovadyahu 1992,


V.Gantmakher et al 1996


T
0
= 3


15 K


Local tunnelling data



B.Sacepe et al 2007
-
8



Nernst effect above T
c



P.Spathis, H.Aubin et al 2008

Phase Diagram

Theoretical model


Simplest BCS attraction model,



but

for critical (or weakly)
localized electrons

H = H
0

-

g

d
3
r

Ψ


Ψ


Ψ

Ψ



Ψ

=
Σ

c
j

Ψ
j

(r)

Basis of localized eigenfunctions

M. Ma and P. Lee (1985) :

S
-
I transition at

δ
L



T
c

Superconductivity at the
Localization Threshold:

δ
L



0

Consider Fermi energy very close

to the mobility edge:

single
-
electron states are extended

but


fractal

and
populate small fraction of the

whole volume

How BCS theory should be modified to account


for eigenstate’s fractality ?

Method: combination of analitic theory and numerical

data for Anderson mobility edge model

Mean
-
Field Eq. for T
c

3D Anderson model:
γ
= 0.57




D
2



1.3 in 3D

Fractality of wavefunctions

IPR:

M
i

=


4

d
r


Modified mean
-
field approximation
for critical temperature T
c

For small


this T
c

is higher than BCS value !

Alternative method to find Tc:

Virial expansion


(A.Larkin & D.Khmelnitsky 1970)


T
c

from 3 different calculations


Modified MFA equation

leads to:

BCS theory:
T
c

=
ω
D

exp(
-
1/
λ
)

Order parameter in real space


for
ξ

=
ξ
k

Fluctuations of SC order parameter


SC fraction =

Higher moments:

prefactor
≈ 1.7 for
γ

= 0.57

With Prob = p << 1
Δ
(r) =
Δ

, otherwise
Δ
(r) =0


Tunnelling DoS

Asymmetry in local DoS:

Average DoS:

Neglected : off
-
diagonal terms

Non
-
pair
-
wise terms with 3 or 4 different eigenstates were omitted

To estimate the accuracy we derived effective Ginzburg
-
Landau functional taking these terms into account

Superconductivity at the

Mobility Edge: major features

-
Critical temperature T
c


is well
-
defined through


the whole

system in spite of strong
Δ
(r)


fluctuations


-
Local DoS strongly fluctuates in real space; it
results in asymmetric tunnel conductance


G(V,r)


G(
-
V,r)

-
Both thermal (Gi) and mesoscopic (Gi
d
)
fluctuational parameters of the GL functional are
of order unity


Superconductivity with Pseudogap


Now we move

Fermi
-

level into the


range of localized eigenstates


Local pairing


in addition to

collective pairing

1. Parity gap in ultrasmall grains



K. Matveev and A. Larkin 1997

No


many
-
body correlations


Local pairing energy

Correlations between pairs of electrons localized in the same “orbital”

-------

-------

E
F

--
↑↓
--

--

 ↓
--

2. Parity gap for Anderson
-
localized eigenstates


Energy of two single
-
particle excitations after depairing:

P(M) distribution


Activation energy T
I

from Shahar
-
Ovadyahu exp. and fit to theory


The fit was obtained with


single fitting parameter


= 0.05

= 400 K

Example of consistent choice:


Critical temperature in the
pseudogap regime


Here we use
M(
ω
)

specific for localized states

MFA is OK as long as


MFA:

is large


Correlation function

M(
ω
)

No saturation at
ω

<
δ
L

:

M(
ω
) ~ ln
2

(
δ
L

/

ω
)

(Cuevas & Kravtsov PRB,2007)


Superconductivity with

Tc <
δ
L

is possible


This region was not found


previously


Here

“local gap”

exceeds
SC gap :

Critical temperature in the
pseudogap regime


We need to estimate

MFA:

It is nearly constant in a


very broad range of


T
c

versus
Pseudogap

Transition exists even at
δ
L

>> T
c0

Virial expansion results:

Single
-
electron states suppressed by pseudogap


Effective number of interacting neighbours

“Pseudospin” approximation

Third Scenario


Bosonic mechanism
: preformed Cooper pairs +
competition Josephson v/s Coulomb


S I T in arrays


Fermionic mechanism
: suppressed Cooper attraction, no
paring


S M T



Pseudospin mechanism:
individually localized pairs



-

S I T in amorphous media



SIT occurs at small Z and lead to paired insulator



How to describe this quantum phase transition ?
Cayley tree model is solved (
L.Ioffe & M.Mezard
)



Qualitative features of

“Pseudogaped Superconductivity”:



STM DoS evolution with T



Double
-
peak structure in point
-
contact
conuctance




Nonconservation of full spectral weight
across T
c




Superconductor
-
Insulator
Transition


Simplified model of competition
between random local energies
(
ξ
i
S
i
z

term) and XY coupling






Phase diagram

Superconductor

Hopping insulator



g

Temperature

Energy

RSB state

Full localization:

Insulator with

Discrete levels

MFA line

g
c



Fixed activation energy is due to the absence of thermal bath at low
ω



Conclusions



Pairing on nearly
-
critical states produces fractal
superconductivity with relatively high T
c
but very small
superconductive density



Pairing of electrons on localized states leads to hard gap


and Arrhenius resistivity for 1e transport



Pseudogap behaviour is generic near


S
-
I transition, with “insulating gap” above T
c




New type of S
-
I phase transition is described


(on Cayley tree, at least). On insulating side activation of
pair

transport

is due to
ManyBodyLocalization
threshold



Coulomb enchancement near mobility edge ??


Condition of universal screening:

Normally, Coulomb interaction is overscreened,
with universal effective coupling constant
~ 1

Example of a
-
InO
x

Effective Couloomb potential is weak:

Class of relevant materials


Amorphously disordered


(no structural grains)


Low carrier density


( around 10
21
cm
-
3
at low temp.)

Examples:


InO
x

NbN
x

thick films or bulk

(+ B
-
doped Diamond?)


TiN

thin films


Be, Bi (
ultra thin films)